Question

State the whether given statement is true or false$ABCD$ is a parallelogram and the bisector of $\angle A$ bisects $BC$ at $x$. Prove that $AD=2AB$.

• A. True
• B. False

Answer 1

The correct option is A True

Let the bisector of $\angle A$ bisects the side $BC$ at $X$.

Given, $ABCD$ is a parallelogram.

∴ $AD||BC$ (Opposite sides of the parallelogram are parallel)

Now, $AD||BC$ and AX is the transversal ,

∴ $\angle 2=\angle 3$ (Alternate angles) ............(1)

and $\angle 1=\angle 2$ ($AX$ is the bisector of $\angle A$) ................(2)

From (1) and (2), we obtain

$\angle 1=\angle 3$

Now, in $\Delta ABX$,

$\angle 1=\angle 3$

⇒ $AB=BX$ ( If two angles of a triangle are equal, then sides opposite to

them are equal)

⇒ $2AB=2BX=BX+BX=BX+XC$ ( $X$ is the mid point of $BC$)

⇒ $2AB=BC$

⇒ $2AB=AD$ (Opposite sides of a parallelogram are equal)

∴ $AD=2AB.$